Question 482538
Given:  {{{sqrt(x+4)+sqrt(x-4)=sqrt(2x+6)}}}
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To solve, square both sides. For the left side this means multiplying:
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{{{(sqrt(x+4)+sqrt(x-4))*(sqrt(x+4)+sqrt(x-4))}}}
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Use the FOIL system. In the two sets of parentheses, multiply FIRST terms, then multiply OUTSIDE terms, then multiply INSIDE terms, and finally multiply LAST terms. 
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Firsts:  {{{(sqrt(x+4))*(sqrt(x+4))= x+4}}}
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Outsides: {{{(sqrt(x+4))*(sqrt(x-4)) = sqrt(x^2 - 16)}}}
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Insides: {{{(sqrt(x-4))*(sqrt(x+4)) = sqrt(x^2 - 16)}}}
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Finally, Lasts: {{{(sqrt(x-4))*(sqrt(x-4))= x-4}}}
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Add all these results together and you have:
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{{{x+4+sqrt(x^2 - 16)+sqrt(x^2 - 16)+x-4}}}
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The two terms containing the radicals are identical. Therefore, they can be added and the overall expression becomes:
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{{{x+4+2*sqrt(x^2 - 16)+x-4}}}
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Note that the +4 and the -4 cancel each other out and also the x + x sum to 2x.  Therefore, the left side that we squared simplifies to:
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{{{2x + 2*sqrt(x^2-16)}}} <--- this is the left side squared
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Now let's return to the original equation and square the right side as follows:
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{{{(sqrt(2x+6))*(sqrt(2x+6))= 2x+6}}}  <--- this is the right side squared
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And the square of the left side must equal the square of the right side as follows:
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{{{2x + 2*sqrt(x^2-16)=2x+6}}}
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Note that a term 2x appears on both sides. Therefore, subtract 2x from both sides and the equation reduces to:
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{{{2*sqrt(x^2-16) = 6}}}
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Divide both sides by 2 and you get:
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{{{sqrt(x^2-16) = 3}}}
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Now get rid of the radical by squaring both sides again and you have
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{{{x^2-16 = 9}}}
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Get rid of the -16 on the left side by adding 16 to both sides and you have:
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{{{x^2 = 25}}}
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And finally, solve for x by taking the square root of both sides. The result is:
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{{{x = 5}}}
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And that's the answer. You can return to the original equation that you were given to solve and substitute 5 for x to get:
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{{{sqrt(x+4)+sqrt(x-4)= sqrt(2x+6)}}}
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Letting x = 5 results in this becoming:
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{{{sqrt(5+4)+sqrt(5-1)= sqrt((2*5)+6)}}}
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and this simplifies to:
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{{{sqrt(9)+sqrt(1)= sqrt(16)}}}
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and finally our check shows that by taking the square roots of the three numbers:
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{{{3+1 = 4}}}
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which means that if x = 5 the original equation balances. Both sides are equal.
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Hope this helps you to understand the problem and how to work it.